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Perturbations of minimizing movements and curves of maximal slope

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  • We modify the De Giorgi's minimizing movements scheme for a functional $φ$, by perturbing the dissipation term, and find a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope for $φ$ with a given rate. We show that if we relax the condition on the perturbations we may have many different meaningful effects; in particular, some perturbed minimizing movements may explore different potential wells.

    Mathematics Subject Classification: 47J35, 47J30, 35B27, 35K90, 49J05.


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  • Figure 1.  Graphs of the discrete solutions with different values of $\tau$. The smaller jumps of $u^\tau$ correspond to the larger parameter $\beta$

    Figure 2.  Pinned motion produced by perturbations diverging in the interval $(1, 2)$

    Figure 3.  The graphs represent two discrete solutions for the same value of $\tau$, corresponding respectively to perturbations as in (25) and (26). Note the discontinuous behavior on the left, while on the right jumps are going to disappear

    Figure 4.  On the left the time chart of $u^\tau$, on the right the plot of $\phi(u^\tau)$, corresponding to perturbations with $\delta = 1$. Note how the motion exits from the lower energy state when $t = 1$

    Figure 5.  Graphs as in Figure 4 with $\delta = 8$. It is grater than the critical value $e^2$, indeed when $t = 1$ the motion does not exit the first well

    Figure 6.  Graphs for $\delta = 8$. As in Figure 5, the motion does not exit the well when $t = 1$, but when $t = 2$ it does

    Figure 7.  Graph for $\delta = 1$. The motion always passes to the very next potential well

    Figure 8.  Graphs of a discrete solution passing through two potential wells at every jump discontinuity

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